Map



Dec..14 ,'1926.

Fig. 1

1,610,413 s. W. BALcH Filed Dec. y12. 1924 l s. W. BALCH MAP 5 SheelgslSh'eet 2 Filed Deo. 12. 1924 y `Inventing wwmwav Dec. 14 1926.

S. rW. BALCH MAP Filed Deo. 12. 1924 5 Sheets-'Sheet 3 Fig. 6

S. W. BALCH Dec. 14 1926.

MAP

Filed Dec. l2. 1924 Inventor,

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Dec. 14,1926. Lwms I s. W. BALCH MAP Filed Dec. 12. 1924 5 Sheyets-lSheet 5 Fig- 9.

GTlQIT-c Proj eckion` I nve ntor- @had WM in part of 'my' application filed Novemberl 13, 1922, Serial/No.` 600,547V for'ships lcourse v .PatentedfDem 14, 192252-` v sAlunEL WQ Baton, cFfMcNTctjAin-NEW JnnsY.

" `MAP;

i Application Yfiled" December rlhisr*application ils-filed asa continuation and 'position1-indicators. fri Y v For the purposes ofnavigation it is` important to have maps-or :chartslon Hat or plane sheets which fulfillthree mathematical conditions, First,

should he'such that it Will berconvenient to draw onthe map the course of the shortest 4sea-levelv line between any two points, and to ascertain the latitude andlongitude at any intermediate pointo'f the course." Such a line is commonly known as the V arcfofagreat Acircle and would be iffthe earth Werey a true sphereybut is infact'a' line v,to be otherwise deiinedlsince the earthy approximates vcloselyjto a spheroid orellips'oid cf revolution, and Will he 1termed a geodesic line. Secondit should be convenient' to as'-y certain theY an'glej at which any geodesic line crosses any 'intermediatel (meridian. Third, it should be convenientpito"ascertain the,A length of'such "a-line and the distance befy tweenjany two points on sucha line.

Assuming that it is desired to ascertain from ay map, if possible,` these three kindsof facts'vvith the aid of the three common draft-fu ingltools,lth e straight e dge,the p rotractor, and the scalerespectively, and further', asfV sumingthat .the surface ofa true sphereis to be represented,.wliich the earthjjis' notya form of map projectionfknown as vthe gnomic has been devised -forthe firstfan'd a .form ofv Y- maplayoutknoivn as the Mercator has been' devised 'for`the second7 AeachhoWever coni-j-V plet'ely sacri'cing the property obtained'by` the other as such maps arei0rdinaril3'f 'inad., As to the third'conditiony'the layout ,of a map on'a pla-ne surface to representfaccord# ingY to v.scale a ylarge area. of Ythe""earth vis deemed impossible? The objectof thisf invention-isto provide a map' on a Asheet which canbe brought into' d i a Aplane?.and which can thereforefbe' printed on a sheet' of paper, and Whichwillpiacti-,-

. cally fullillall three of theseV conditions and allow for the spheroidal figure o f theearth.

For use ivitlithe highest degreeof accuracy i of which the map is capablefitfis assumed that the instrument set', `forth in my above mentioned application is to be utilized' `for, the three functions ofn thestraight.I edge, pro-` tractor and"scale, but litf'is also intended. to provide maps from' u'hiclisuchfacts canbe.

thel gconstruction gascertainedrvith;increasedfaccuracy hy the.

aid'ofl thecrdinary drafting tools."

c Figure@4 shows the spheroid of reference used inVA kclosely"approximates Y {Fig-f2 shoivs sphereto which Ta section lof the spheroidal'surface can be conformed Without material distortion. A

In the accompanying'ive"sheetsof dran? 1 ings which forni fa part Aof this .de'scripti'o'n,

geodet-ic surveying' to which the earth" L '6K5 1v 3 illustrates theprojection Vfrom this Vconformalsphere to a tangentcylinder. Fig. 4: illustrates the projection froml the conformal sphere 4 to gnonnc mapx lFig. shoivs the sphericaltrianglcs--int Y* volved in calculatingH the; coordinates 'for' laf t tangent cylinder projection.l

inapembodying'thisginvention. f v

invention.

A `general cator, or gnomic principlesffronithe spheroid a Y, tangent..V Plane; 'fon a a rig.- e Shows'amspherical','efiangis volved inicalculating :the j'coordina'tes ,'f'rja Figa?? `shows adevelopedtaiagentlcylinder` j y Fig'. S'shows afcom'parison cfordinates for a: tangent cylinder {md-Harlilercatorf.map.'`

Fig- 9 shows a .gnomic mapembodying: this .l rulefforgtlie maps on .elther thev tangentcylinder,'Mer-f of reference has heretofore been #deemed to Y involve insuperabie mathematicalditlculties,V i 1 and suchrules asjliave Ibeengfii'zen, have genl erallybeen Vfor projections from a spheref *90 of the earth, except when the equator'is made "Without allowance V for the' spheroidal form the axisV of Athe inap.

As preliminary step to "the: practicalgV laying'out offfa map according toleither off` lthe above mentioned principles `a sphere is sought'to 4which thefdesired portion lof the spheroidalj surface. can he, conformed vhy bending and Withcut material' alteration off dimensions. `The spheroid of reference AhasV ain Aequatorial radius of 6,378,206"kilometers and a polar( radiusjcf f/kilometers.

The drawing'is. made to a greater eccentricity than they dimensicns indicate so that the --si effect may beapparent Without applying in` s y tudes sixty north and south and --is not a' complete endless hand, then itY can benonorrned'to sphere vwith a radius Rfequal's 6.361.894I kilometers Without` material distortion. f A minute or are; Y(in/this conformal Vsphere has a length ot lOliloinetersand Yolf 1.1499 statute iniles. lf this surta'cel'whieh is transferred is .assumed to carry with Yit theg-graticule of ineridians and parallels at even tenfdegree interv'alsfasV sho'Wn'in full lines, these lines 'gwill not' coincide With the corres Dondin meridiane and arallels Y w 1 of the conrorinal spnere Shown as dotted lines, but they parallels Vwill be shiftedtonard i theequator by Various amounts around fsii; lneinut'es'V otfarc and the` ineridians will be spread' apart nine seconds to a degree.

the following table `is shown the conformal or reducedA latitude 6b.' andthe conformal, or reduced longitude A, the first column' being thegoriginal sphero'idaly latitude'and longitude.v or' Qv'lliereduction lfor any intermediate latitude C'anfb'e'ound by interpolation. The rey1duct ion-for longitude Consists inadding nine seeondsV for each" degree and may be en; tended. beyond Whatj-theftableshows if necessary. Vlhe operation of making aniap f consists 1 in 'transferring the lgraticule troni VVthe sphe'roid of reference toA the conforma sphere, projecting the transformed graticule Yaccording to any 'of' the established methods of inap projeotionfand Ythen restoring vthe original even degree :designations tov the lneridians and parallels of the projeeted grati- The inethods ot projection or construction from a sphere 'will he shownfor three types 'of maps in which theincreeljsed`v accuracy rresultingroni the foregoing transformations Ycanfhe' utilized.

The vmagent-og//Zlz'naler mdp;

The object'heing vprirnarily toproduce a Y* map of serviceinnavigation,there exist-sA for it line 'of especial' interest which is the Course oitravel. Let Gle vhe the'line ot interest tangent to `'oonforinal latitude andV crossing the equator at anglel yat` a point stantY ninety' degreesv `from the point fof'tanf Y genojy to the' parallel. 1 AThe projection is de; to aljeylinder'tangentf'along*thisline kof interest. lnthediagrani, Fig.' 5 the conformal sphere issiiownwith EF the equator, the poleandOFan are of the great circle along which thefcylind-erisf tangenti". Coordinates from@ oi VVthe point Pon they cylinder longitude )tand latitude 11) are found tollows. The coordinates onthe Vsphere will be designatedasV ares as and y, and it Y a is inrutes elevare, the coordinateson'.`

the cylinder inlrilonieters willbe 1.850@ a' anelli vtan 7J'. The following equations can be rritten *for s aherical right angle triangles;

There. is saine sign fon'pas for Q, and opl posite sign for y.

v'llo villustrate theA application ofthetorl V.inulae let itloe yrequired to find the coordinates` for. the gratioule,` on aeylnder tangent` along a'great. ycirole at inclination yv equals 50, .the calculationheing .for the point l? lnrfthivs partieular'exainple @and g are niinus quantities. l Hence is less VKthan p plus are Yto he regarded algebraieally. Y

, .71 :lung

Y -172 337224Y Y `50 60 00 log:

f The quantities, a; are q-l-Qylog. eros" ,8,15 'l Y Y and logiR tan are' the same for additional.; points onthe saine nieridan and the calculaV tion of' they Y coordinates' isv g Vaccordingly shortened. l

@Having calculated the coordinates/toi thev east n ofthe origin they vcan be use'lfor plotthe sign of wf'and for thesouthernhemispherebyl reversingfthe signs vof y. Ying the lorigin at longitudeBOO West, eXtjend ing the'aXis forty degrees yWestward .and twenty-'degrees eastward, and mapping'ten degrees each sideof the axis,- allof the sailtangent-cylinder projection.

error. Y

' ...The Mercatrmapj l The Mercator map has an axis like the As generally known, the equator is the axis. LV.To plot such a map from the conformal sphere on an oblique axis the abscissa for any point is the same as forthe tangent cylinder projection.

Y The ordinates are slightly shorter,1 the dif- Yis added to the constant.

feren'ce'being slight forthe first ten degrees, Aas Will be seen from Fig. 8 in which the ordinates up to ten degrees are plotted tothe same scale for each degree on b oth systems for comparison. To compute'the length of g/ is rst found by the folan ordinate, arc lowing equation:

3.898489-l-log. [log cot 1/2 (90-y)] To find the length of an ordinate VWith thea kilometer as the unit, the logarithm of 1.8506

ample the Work is as vfollows By -plac In the above eX- jection by prolonging radii from,l intersec-- ftincf olnts of/the transferred Oraticulefon` o P b the conformalA sphere toafplane'G, H, 1,13

meridian'.` Coordinates w',

y 'of a point P are found fromthe longitude To illustrate the application of the formu- `which is tangent at T, latitude'y on the zero;

`and latitude y by. the following equatitms'z,V lng courses betweennorthern American andA northern European. points on Ythefrttlantic are included. By thus restricting, the Width ofthe map, serious distortion is avoided7 and for most'purposes all of the above mentioned sets offacts can be obtained from itvvith or- .l dinary drafting tools and, with negligible y 15 Y f 99 tudev y equals 45, the calculation being for Y,

the point P at-'geodeticlongitude 40,'(conformallongitude )t:400600), and geodetic latitude 25, =2454^40).

oll/

10g. con 0.000000 10g. cos 9. 883017 10g. cot 9. 883617 log tan 0. 000000 log. cos 9. 883617 log. tan 9. 883617 10g eos 9. 947233 colog. sin 0. 171542 10g. cot 0.083120 Constfmf M 1og.1og. cot-2q. 919702 Y AV map plotted from coordinates calculated by either ofthe above methods With a scale having one-tenth of a millimeter as'its unit will show the `earth7s surface to ya scale of one to ten `million with negligible error for several degrees on eitherside of the axis.-` The gnomom'c vmap. l The gnomonic map is illustrated as apro- I 10g; cot v"0.074649 lo'g. 0.240191 10g. sin 9.038379 4 10g. sin 9. 335407 -12 30 07 10g. sin 9. 335407 VIn plottingitwill bevconvenient to start with; a sheet even if the portion adjacent to itis to be discarded in the finished map. The position of; the poleis first' determined. Bot-l1 coordinates are. calculated for only one point on each meridian along the edge of the inap remotefrom the pol 'These pointsland the pole define all of the meridians since theyvare straight'linesin'this projection. Additional points on a meridian can be determined from Y (conformal latitude iog sin 9. 808961 large enough to Vinclude the pole Vthe ordinate only. TheY quant-ities, log u, a, and jare the, same for 'addlt-onal pomtsl- ,on the samefln-erldlanzand do notvhave'to be recalculated f I olzurn: I

111A map having:agratolowlclris Con-ff' Y structed from@ (1o13j1"rrml jsph-ere\nfwheh; each parallel of latitude-yhasv boendjsplaoed toward the equator andthe' rnerdgnsspread "bye an amount which" will bring Vdistances by an` amountwhch "Willbrngjdtances Valong the lnerdiansi and parallels ntofgub# stantally the Sarnesoale, the map havngjanaxis-to Vanfrololque great circle oflthe'eon-- .ormals'phere Y jevoted to 'a tangent ylnderl fromnf Con-1 'olmalfspleren which each parallel' offlattude .hasV been displaced toward the, eqfutor andA the Iner-di-n's spread by an' amount which will bring distances along the mef ridiamo-'and parallels into-substantially tlief samesoale, theta-ugentagijisvbeng anioblique kgreat'circle of? the conformalsphere. l

SMUEL ,Wyv BALGH. 

